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A356483
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a(n) is the hafnian of a symmetric Toeplitz matrix M(2*n) whose first row consists of prime(1), prime(2), ..., prime(2*n).
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9
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1, 3, 55, 2999, 347391, 69702479, 22441691645, 10776262328919, 7190279422736061, 6439969796874334809, 7447188585071730451961
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OFFSET
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0,2
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LINKS
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EXAMPLE
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a(2) = 55 because the hafnian of
2 3 5 7
3 2 3 5
5 3 2 3
7 5 3 2
equals M_{1,2}*M_{3,4} + M_{1,3}*M_{2,4} + M_{1,4}*M_{2,3} = 55.
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MATHEMATICA
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k[i_]:=Prime[i]; M[i_, j_, n_]:=Part[Part[ToeplitzMatrix[Array[k, n]], i], j]; a[n_]:=Sum[Product[M[Part[PermutationList[s, 2n], 2i-1], Part[PermutationList[s, 2n], 2i], 2n], {i, n}], {s, SymmetricGroup[2n]//GroupElements}]/(n!*2^n); Array[a, 6, 0]
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PROG
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(PARI) tm(n) = my(m = matrix(n, n, i, j, if (i==1, prime(j), if (j==1, prime(i))))); for (i=2, n, for (j=2, n, m[i, j] = m[i-1, j-1]; ); ); m;
a(n) = my(m = tm(2*n), s=0); forperm([1..2*n], p, s += prod(j=1, n, m[p[2*j-1], p[2*j]]); ); s/(n!*2^n); \\ Michel Marcus, May 02 2023
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CROSSREFS
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KEYWORD
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nonn,hard,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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